Large time behavior of the logistic age-structured population model in a changing environment

Abstract

Population growth is governed by many external and internal factors. In order to study their effects on population dynamics, we develop an age-structured time-dependent population model with logistic-type nonlinearity. We prove existence of a unique nonnegative bounded solution. Our main concern is to study asymptotic behavior of a solution in the general case, and especially for a periodic environment. We use the method of lower and upper solutions known in the theory of integral equations to formulate lower and upper boundaries of population density. In the periodic case, we discover a connection between the period of oscillation and its effect on population growth.

Density–Dependent Feedback in Age–Structured Populations

“…Now note that equation (18) together with the fact β is bounded implies that the number of newborns satisfies the upper estimate in (35). Since P is bounded and β is bounded from below on (b 1 , b 2 ) we have that the left-hand side of (35) is true as well.…”

“…Assumption (A 1 ) corresponds to the fact that for large populations mortality is increased by increase in population size and also generalizes mortality rate used in [18]. Note that for small populations this correlation does not need to hold.…”

“…Mathematical representation of the negative-density dependence begins with the Verhulst model for unstructured population, see for example [15], and the consequence of this type of regulation are bounded growth and stabilization of population around its carrying capacity. Effects of the negative densitydependence on the age-structured population are studied in [18]. Under the assumption that only members of the same age-class compete, the existence of a bounded solution has been proven.…”

“…Using the framework proposed in [16], we improve our previous assumptions used in [18], [19] that intraspecific competition occurs only among individuals of the same age by using more realistic age, and the density-dependent mortality µ(a, P (t)) and the fertility β(a, Q(t)), where P (t) = ∞ 0 p(a)n(a, t) da, Q(t) = ∞ 0 q(a)n(a, t) da are weighted populations. Here n(a, t) denotes the number of individuals of age a at time t and p(a), q(a) are certain weight functions.…”

“…Mathematical models of age structured populations usually use density dependent vital rates without any special regard to the type of feedback that densitydependence produces; see for example [2,3,10,11,13,21]. On the other hand, some authors investigate consequences of the Allee effect in age-structured populations, see for instance [7,8], or intraspecific competition [18].…”